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Theorem tgcgrtriv 25092
Description: Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgcgrtriv.1 (𝜑𝐴𝑃)
tgcgrtriv.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgcgrtriv (𝜑 → (𝐴 𝐴) = (𝐵 𝐵))

Proof of Theorem tgcgrtriv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . . 5 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . . 5 = (dist‘𝐺)
3 tkgeom.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 757 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐺 ∈ TarskiG)
6 tgcgrtriv.1 . . . . . 6 (𝜑𝐴𝑃)
76ad2antrr 757 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐴𝑃)
8 simplr 787 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝑥𝑃)
9 tgcgrtriv.2 . . . . . 6 (𝜑𝐵𝑃)
109ad2antrr 757 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐵𝑃)
11 simprr 791 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝑥) = (𝐵 𝐵))
121, 2, 3, 5, 7, 8, 10, 11axtgcgrid 25075 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → 𝐴 = 𝑥)
1312oveq2d 6539 . . 3 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝐴) = (𝐴 𝑥))
1413, 11eqtrd 2639 . 2 (((𝜑𝑥𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵))) → (𝐴 𝐴) = (𝐵 𝐵))
151, 2, 3, 4, 9, 6, 9, 9axtgsegcon 25076 . 2 (𝜑 → ∃𝑥𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 𝑥) = (𝐵 𝐵)))
1614, 15r19.29a 3055 1 (𝜑 → (𝐴 𝐴) = (𝐵 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  cfv 5786  (class class class)co 6523  Basecbs 15637  distcds 15719  TarskiGcstrkg 25042  Itvcitv 25048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-nul 4708
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-ov 6526  df-trkgc 25060  df-trkgcb 25062  df-trkg 25065
This theorem is referenced by:  tgcgrextend  25093  tgcgrsub  25118  iscgrglt  25123  trgcgrg  25124  tgbtwnconn1lem3  25183  leg0  25201
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